Progress would be exceedingly slow, but your people can somehow remember skills like reading and writing, which can serve as a sort of substitute memory once they develop it. They would probably have to be obsessive at noting things down, or else spend a lot of time - a lot more extra time going over the same ground day after day because they forgot they had already done it. And they would lose the thinking time - noting down ideas will help, but following a train of thought would be difficult over time with only written records, and not time to let ideas "stew" inside the mind to synthesize.
On the other hand, learning actual skills is the only way for this people to advance themselves, they can't lean on rote memorization (as our education systems sometimes do) - so perhaps they will get the most out of their education, when they can get it, instead of memorize-and-forgetting it. they would have to learn mathematical formulas, instead of memorizing constants, and learn critical thinking and analysis, instead of memorizing facts and 'standard' interpretations. The line between "learnable skill" and "personal memory" is going to be really, really important to what they can learn, and what they forget.
But, given that they might end up being obsessive over memory, possibly their technology would revolve around overcoming this handicap - which could let them reach an advanced civilization anyway. (similarly, perhaps, to the way so much of our technology revolves around our lack of physical weaponry, so we're always searching for ways to be more and more lethal). This civilization would evolve writing, drawing, and storing records to the most efficient levels. The development of photographs and video would also be prized, and encouraged to develop early and deeply to serve as a prosthetic memory.
Their cave drawings, rather than being symbolic or religious (or historical) would have crude maps to the best places, and a cruder tallying of how often they go to any place, so they know what's picked over, and what's still likely to have food. They would use tokens, maybe, to show what had been done or said or decided on any day, that needed to be acted on the next (like who did which chore, who slacked off, who needed punished). They would develop writing, and ways of preserving their records, as elaborately as they could - because it is instant value, and the more information they can pack in there the better.
It's still a pretty big handicap to overcome, though.